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arXiv:hep-th/0309105 · International Journal of Modern Physics A

TWO COUPLED HARMONIC OSCILLATORS ON NONCOMMUTATIVE PLANE

Ahmed Jellal, MICHAEL SCHREIBER, El Hassan El Kinani

Published Mar 20, 2005• 24 citations
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Abstract

We investigate a system of two coupled harmonic oscillators on the noncommutative plane [Formula: see text] by requiring that the spatial coordinates do not commute. We show that the system can be diagonalized by a suitable transformation, i.e. a rotation with a mixing angle α. The obtained eigenstates as well as the eigenvalues depend on the noncommutativity parameter θ. Focusing on the ground state wave function before the transformation, we calculate the density matrix ρ0(θ) and find that its traces Tr (ρ0(θ)) and [Formula: see text] are not affected by the noncommutativity. Evaluating the Wigner function on [Formula: see text] confirms this. The uncertainty relation is explicitly determined and found to depend on θ. For small values of θ, the relation is shifted by a θ2 term, which can be interpreted as a noncommutativity correction. The calculated entropy does not change with respect to the normal case. We consider the limits α = 0 and α = π/2. In the first case, by identifying θ to the squared magnetic length, one can recover basic features of the Hall system.

DOI
10.1142/s0217751x05020835

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We investigate a system of two coupled harmonic oscillators on the noncommutative plane [Formula: see text] by requiring that the spatial coordinates do not commute. We show that the system can be diagonalized by a suitable transformation, i.e. a rotation with a mixing angle α. The obtained eigenstates as well as the eigenvalues depend on the noncommutativity parameter θ. Focusing on the ground state wave function before the transformation, we calculate the density matrix ρ0(θ) and find that its traces Tr (ρ0(θ)) and [Formula: see text] are not affected by the noncommutativity. Evaluating the Wigner function on [Formula: see text] confirms this. The uncertainty relation is explicitly determined and found to depend on θ. For small values of θ, the relation is shifted by a θ2 term, which can be interpreted as a noncommutativity correction. The calculated entropy does not change with respect to the normal case. We consider the limits α = 0 and α = π/2. In the first case, by identifying θ to the squared magnetic length, one can recover basic features of the Hall system.
We investigate a system of two coupled harmonic oscillators on the non-commutative plane \\RR^2_{\\theta} by requiring that the spatial coordinates do not commute. We show that the system can be diagonalized by a suitable transformation, i.e. a rotation with a mixing angle \\alpha. The obtained eigenstates as well as the eigenvalues depend on the non-commutativity parameter \\theta. Focusing on the ground state wave function before the transformation, we calculate the density matrix \\rho_0(\\theta) and find that its traces {\\rm Tr}(\\rho_{0}(\\theta)) and {\\rm Tr}(\\rho_0^2(\\theta)) are not affected by the non-commutativity. Evaluating the Wigner function on \\RR^2_{\\theta} confirms this. The uncertainty relation is explicitly determined and found to depend on \\theta. For small values of \\theta, the relation is shifted by a \\theta^2 term, which can be interpreted as a quantum correction. The calculated entropy does not change with respect to the normal case. We consider the limits \\alpha=1 and \\alpha={\\pi\\over 2}. In first case, by identifying \\theta to the squared magnetic length, one can recover basic features of the Hall system.
The thermodynamical properties of a system of two coupled harmonic oscillators in the presence of a uniform magnetic field B are investigated. Using an unitary transformation, we show that the system can be diagonalized in a simple way and then obtain the energy spectrum solutions. These are then used to determine the thermodynamical potential in terms of different physical parameters like the coupling parameter α. This allows us to give a generalization of significant work already published and obtain different results, which can be used to discuss the magnetism of the system. Different limiting cases, in terms of α and B, are discussed. In fact, quantum corrections to the Landau diamagnetism and orbital paramagnetism are found.
The quantum entanglement of two connected harmonic oscillators via angular momentum yielding a magnetic coupling [Formula: see text] is discussed in this study. The corresponding Hamiltonian is diagonalized by using three canonical transformations, and then the stationary wave function is obtained. The Schmidt decomposition is used to explicitly determine the modes [Formula: see text], with [Formula: see text], [Formula: see text] and [Formula: see text] being two quantum numbers associated with the two oscillators. We summarize our findings by looking at the effects of anisotropy [Formula: see text], [Formula: see text], asymmetry [Formula: see text], and dynamics on entanglement. (i) With increasing [Formula: see text], the entanglement grows exceedingly large. (ii) The sensitivity to [Formula: see text] is determined by [Formula: see text] and [Formula: see text]. The physical parameters and quantum numbers play a great role in the periodic resuscitation of entanglement.
In order to evaluate the Feynman path integral in noncommutative quantum mechanics, we consider properties of a Lagrangian related to a quadratic Hamiltonian with noncommutative spatial coordinates. A quantum-mechanical system with noncommutative coordinates is equivalent to another one with commutative coordinates. We found connection between quadratic classical Lagrangians of these two systems. We also shown that there is a subclass of quadratic Lagrangians, which includes harmonic oscillator and particle in a constant field, whose connection between ordinary and noncommutative regimes can be expressed as a linear change of position in terms of a new position and velocity.